Shehu transform

In mathematics, the Shehu transform is an integral transform which generalizes both the Laplace transform and the Sumudu integral transform. It was introduced by Shehu Maitama and Weidong Zhao^{[1] [2] [3]} in 2019 and applied to both ordinary and partial differential equations.^{[4] [3] [5] [6] [7] [8]}

The Shehu transform of a function ${\displaystyle f(t)}${\displaystyle f(t)} is defined over the set of functions

${\displaystyle A=\{f(t):\exists M,p_{1},p_{2}>0,|f(t)|<M\exp(|t|/p_{i}),,円,円,円{\text{if}},円,円,円t\in (-1)^{i}\times [0,,円\infty )\}}${\displaystyle A=\{f(t):\exists M,p_{1},p_{2}>0,|f(t)|<M\exp(|t|/p_{i}),,円,円,円{\text{if}},円,円,円t\in (-1)^{i}\times [0,,円\infty )\}}

as

${\displaystyle \mathbb {S} [f(t)]=F(s,u)=\int _{0}^{\infty }\exp \left(-{\frac {st}{u}}\right)f(t),円dt=\lim _{\alpha \rightarrow \infty }\int _{0}^{\alpha }\exp \left(-{\frac {st}{u}}\right)f(t),円dt,,円s>0,,円u>0,,円,円,円,円(1)}${\displaystyle \mathbb {S} [f(t)]=F(s,u)=\int _{0}^{\infty }\exp \left(-{\frac {st}{u}}\right)f(t),円dt=\lim _{\alpha \rightarrow \infty }\int _{0}^{\alpha }\exp \left(-{\frac {st}{u}}\right)f(t),円dt,,円s>0,,円u>0,,円,円,円,円(1)}

where ${\displaystyle s}${\displaystyle s} and ${\displaystyle u}${\displaystyle u} are the Shehu transform variables.^[1] The Shehu transform converges to Laplace transform when the variable ${\displaystyle u=1}${\displaystyle u=1}.

The inverse Shehu transform of the function ${\displaystyle f(t)}${\displaystyle f(t)} is defined as

${\displaystyle f(t)=\mathbb {S} ^{-1}[F(s,u)]=\lim _{\beta \rightarrow \infty }{\frac {1}{2\pi i}}\int _{\alpha -i\beta }^{\alpha +i\beta }{\frac {1}{u}}\exp \left({\frac {st}{u}}\right)F(s,u)ds,,円,円,円,円(2)}${\displaystyle f(t)=\mathbb {S} ^{-1}[F(s,u)]=\lim _{\beta \rightarrow \infty }{\frac {1}{2\pi i}}\int _{\alpha -i\beta }^{\alpha +i\beta }{\frac {1}{u}}\exp \left({\frac {st}{u}}\right)F(s,u)ds,,円,円,円,円(2)}

where ${\displaystyle s}${\displaystyle s} is a complex number and ${\displaystyle \alpha }${\displaystyle \alpha } is a real number.^[1]

${\displaystyle {\mathbb {S} }\left[\int _{0}^{t}f(\zeta )d\zeta \right]={\frac {u}{s}}F(s,u),}${\displaystyle {\mathbb {S} }\left[\int _{0}^{t}f(\zeta )d\zeta \right]={\frac {u}{s}}F(s,u),}

where ${\displaystyle {\mathbb {S} }\left[f(\zeta )\right]=F(s,u)}${\displaystyle {\mathbb {S} }\left[f(\zeta )\right]=F(s,u)} and ${\displaystyle f(\zeta )\in A.}${\displaystyle f(\zeta )\in A.}^{[1] [3]}

If the function ${\displaystyle f^{(n)}(t)}${\displaystyle f^{(n)}(t)} is the nth derivative of the function ${\displaystyle f(t)\in A}${\displaystyle f(t)\in A} with respect to ${\displaystyle t}${\displaystyle t}, then ${\displaystyle {\mathbb {S} }\left[f^{(n)}(t)\right]=\left({\frac {s}{u}}\right)^{n}F(s,u)-\sum _{k=0}^{n-1}\left({\frac {s}{u}}\right)^{n-(k+1)}f^{(k)}(0).}${\displaystyle {\mathbb {S} }\left[f^{(n)}(t)\right]=\left({\frac {s}{u}}\right)^{n}F(s,u)-\sum _{k=0}^{n-1}\left({\frac {s}{u}}\right)^{n-(k+1)}f^{(k)}(0).}^{[1] [3]}

Let the functions ${\displaystyle f(t)}${\displaystyle f(t)} and ${\displaystyle g(t)}${\displaystyle g(t)} be in set A. If ${\displaystyle F(s,u)}${\displaystyle F(s,u)} and ${\displaystyle G(s,u)}${\displaystyle G(s,u)} are the Shehu transforms of the functions ${\displaystyle f(t)}${\displaystyle f(t)} and ${\displaystyle g(t)}${\displaystyle g(t)} respectively. Then

${\displaystyle {\mathbb {S} }\left[(f*g)(t)\right]=F(s,u)G(s,u).}${\displaystyle {\mathbb {S} }\left[(f*g)(t)\right]=F(s,u)G(s,u).}

Where ${\displaystyle f*g}${\displaystyle f*g} is the convolution of two functions ${\displaystyle f(t)}${\displaystyle f(t)} and ${\displaystyle g(t)}${\displaystyle g(t)} which is defined as

${\displaystyle \int _{0}^{t}f(\tau )g(t-\tau )d\tau =\int _{0}^{t}f(t-\tau )g(\tau )d\tau .}${\displaystyle \int _{0}^{t}f(\tau )g(t-\tau )d\tau =\int _{0}^{t}f(t-\tau )g(\tau )d\tau .}^{[1] [3]}

• Integral transforms
• Differential equations
• Mathematics stubs

• Articles with short description
• Short description matches Wikidata
• Wikipedia articles with possible conflicts of interest from June 2025
• All stub articles